Prerequisites for reliable modeling with first-principles methods P. Kratzer Fritz-Haber-Institut der MPG D-14195 Berlin-Dahlem, Germany
Prerequisites for modeling (I) Issues to consider when applying Density Functional Theory Is Density Functional Theory (with approximate E xc ) appropriate for the problem under study? DFT has known problems with systems having strongly correlated (e.g. NiO, CoO, ) and/or narrow bands (e.g. arising from partially filled f-shell) Which (approximate) E xc is most suitable? check by comparing to wavefunction-based methods (CC, QMC, ) Is the use of pseudopotentials appropriate, or should one do an allelectron calculation? check by comparing to all-electron calculations for simple bulk structures, often available from the literature
Prerequisites for modeling (II) Issues to consider when applying the plane wave/supercell method How appropriate is the model for your physical system? e.g. cluster versus slab model for surfaces Is your model system large enough to avoid spurious interactions? e.g. slab thickness, size of the unit cell Are there long-range (electrostatic or elastic) interactions? e.g. charged defects in semiconductors, dislocation lines Is the k-point sampling appropriate to describe the dispersion of the bands and the Fermi surface? Is the calculation converged with respect to technical parameters, like cut-off, T el,?
Absolute versus relative convergence Example 1: absolute convergence for GaAs, LDA, Hamann pseudopotential, data points at E cut = 7, 8, 10, 20 Ry empirical dependence on the number of plane waves: E(N) E conv a 1 exp( a 2 N)
The G+k plane-wave basis set spheres with radius (E cut ) 1/2 with origin in k: N ideal = (V/6π 2) (E cut ) 3/2 = 84.941 E cut = 7 Ry V = 271.6 bohr 3 2x2x2 weight plane waves 1 0.25 83 2 0.75 89 weighted average 87.461 4x4x4 weight plane waves 1 0.03125 77 2 0.09375 84 3 0.09375 83 4 0.09375 81 5 0.03125 83 6 0.09375 82 7 0.09375 87 8 0.09375 87 9 0.18750 86 10 0.18750 89 weighted average 85.01
Correction for the discreteness of the basis set The calculated curve must show discontinuities because the N av values are scattered around N ideal. Correcting for this effect gives much better agreement with the equation-of-state curve. Rignanese et al., Phys. Rev. B 52, 8160 (1995); Francis and Payne, J. Phys. Cond. Mat., 17, 1643 (1990) E = N ideal ln(n av /N ideal ) ( E tot (N)/ N) Nideal N ideal ln(n av /N ideal ) a 1 a 2 exp( a 2 N ideal ) Example: GaAs bulk, 7 Ry cut-off, 2 x 2 x 2 k-points
Relative versus absolute convergence Example 2: Si cohesive energy [ev] cohesive energy of bulk Si cutoff energy [Ry] total energy convergence shown on the same scale relative energies converge much faster with the plane wave cutoff than the absolute total energy! A. Ramstad, G. Brocks, and P. J. Kelly, Phys. Rev. B 51, 14504 (1995).
k-points convergence for metals: Pd For a good description of a metal, in particular the bulk modulus, it may be necessary to sample the Fermi surface in great detail! Netscape Hypertext Document http://www.phy.tu-dresden.de/~fermisur/ k-point set 6x6x6 10x10x10 14x14x14 lattice constant [Å] 3.83 3.85 3.85 bulk modulus [Mbar] 2.32 2.26 2.23 B' 5.66 5.63 5.62 cohesive energy [ev] 5.09 5.08 5.06 LDA, LAPW calculation, r MT =2.4, E cut =14 Ry, l max =10 calculations: Juarez da Silva, unpublished
Topic: Surface Chemistry How well are (small) molecules described in DFT? dissociation energies in ev HF LSD GGA(PBE) meta-gga[1] exp. H 2 3.56 4.91 4.53 4.96 4.75 H 2 0 6.27 11.56 10.16 9.98 10.07 HF 3.80 7.04 6.16 6.01 6.10 CO 7.43 12.96 11.65 11.10 11.24 N 2 4.70 11.60 10.55 9.94 9.91 F 2 <0 3.39 2.32 1.87 1.67 O 2 1.36 7.58 6.23 5.70 5.22 [1] J.Perdew et al, Phys. Rev. Lett. 82 (1999) 2544 How well do we describe the surface structure? Can we identify reaction mechanisms and locate transition states for simple surface reactions (e.g. dissociation of a diatomic molecule)?
Surface relaxation: GaAs(001)(2x4) P. Kratzer, unpublished LDA, 10 Ry plane-wave cut-off, 2x4 k points in BZ Slabs with n layers Ebulk(n) := [Esl(n) Esl(n 2)]/Nat surface energy γ(n)a := Esl(n) (NAs+NGa) Ebulk(n) (NAs-NGa)µAs
Quantum size effects: Al(110) E F LDA, 20Ry plane wave cutoff, 16 k-points in IBZ A. Kiejna, J. Peisert and P. Scharoch, Surf. Sci. 432 (1999) 54
Surface reconstruction of Si(001): (2x1) [001] reconstruction by surface dimerization side view symmetric dimers? Jahn-Teller like effect may lift the overlap of surface states. See: J. Dabrowski and M. Scheffler, Appl. Surf. Sci. 56-58, 15 (1992) buckled dimers
Surface reconstruction: Si(001) dimerization energy energy gain from buckling A. Ramstad et al., Phys. Rev. B 51, 14504 (1995). cutoff energy [Ry] cutoff energy [Ry]
Cluster models for the Si(001) surface Is it possible to describe the buckling of the surface Si dimers using a cluster models of the Si (001) surface? angle [deg.] bond length E [ev] HOMO-LUMO [Å] gap[ev] Si 9 6.9 2.21 0.00 1.18 Si 15 15.7 2.28 0.12 0.86 Si 21 18.6 2.35 0.17 0.70 slab 18.9 2.36 0.20 0.00 Si 9 H 12 Si 15 H 16 The buckling only develops fully in a cluster containing 3 surface Si dimers. The failure to describe the buckling is related to differences in the electronic structure. E. Penev, P. Kratzer and M. Scheffler, J. Chem. Phys. 110, 3986 (1999) Si 21 H 20
Electronic correlation effects at the surface Since the symmetric dimer is a bi-radical, the buckling effect could be sensitive to subtle electronic correlation effects. energy gain (mev) 250 200 150 100 50 0 slab result PW91 QMC 9 15 21 27 cluster size (Si atoms) S. B. Healy, C. Filippi, P. Kratzer, E. Penev, M. Scheffler, Phys. Rev. Lett. 87, 016105 (2001)
Arrangement of neighboring Si dimers top view of Si(001)
Periodicity of the reconstruction: effects of k-point sampling Number of k-points A. Ramstad, G. Brocks, and P. J. Kelly, Phys. Rev. B 51, 14504 (1995).
The reaction of H 2 with the Si(001) surface p(2x2) or c(4x2) reconstruction top view cluster model silicon monohydride unit cell (slab) buckled Si dimers (one dangling bond per Si atom) side view terminating (pseudo) hydrogens
H 2 /Si(001): Cluster size convergence Si9 Si41 Si21 desorption energy [kcal/mol] Si89 Si9 Si21 Si41 Si89 slab J. A. Stechel, T. Phung, K. D. Jordan, and P. Nachtigall, J. Phys. Chem. B 105, 4031 (2001).
H 2 /Si(001): Performance of density functionals Both the reaction energy and the barriers are underestimated by the PW91 functional. B3LYP comes close to the QMC results, but still gives slightly lower values. reaction energy (ev) 2.8 2.6 2.4 2.2 2 1.8 9 15 21 27 33 LDA PW91 B3LYP QMC desorption barrier (ev) 4 3.5 3 2.5 2 9 15 21 27 33 cluster size (Si atoms) cluster size (Si atoms) C. Filippi, S. B. Healy, P. Kratzer, E. Pehlke, M. Scheffler, Phys. Rev. Lett. 89, 166102 (2002)
Cu(111): H 2 dissociative adsorption 1.1Å atoms in cell 3 3 4 layers 2 4 6 1.1Å k-points 1 dissociation barrier [ev] 5.48 6 1.41 18 0.35 0.73 0.48 54 0.62 0.54 0.55 162 0.61 0.60 0.51 Note: LDA yields zero dissociation barrier GGA(PW91), Troullier-Martins pseudopotential, 50 Ry plane wave cut-off, P. Kratzer, B. Hammer and J.K. Nørskov, Surf. Sci. 359, 45 (1996)
Importance of surface relaxations Example: CH 4 / Ir(111) slab calculations Strong outward relaxation of the surface Ir atom at the transition state! G. Henkelman and H. Jonsson, Phys. Rev. Lett. 86 (2001), 664
Conclusions The quality of the Brillouin zone sampling ought to be tested carefully for each system, in particular for metals. If long-range effects are to be expected, it can be helpful to explore their size in advance, by using simple (e.g. elastic) models. (meta-)ggas have brought us a good way closer to chemical accuracy for surface reactions. In the DFT approach, slabs are usually a more efficient model of the surface then clusters. When trying to model single isolated objects (e.g. defects, adsorbates ), one needs to be aware of unwanted substratemediated interactions.
A few words of guidance When starting a new project, first repeat some bulk calculations for the chemical elements constituting your material under study. Compare to all-electron calculations, if available, to check the performance of the pseudopotentials! Design your project: For each critical parameter, find out a satisfactory production setting and an improved check setting. Make sure that the checks are feasible with your computer hardware. Complete all checks before going into the production phase. When producing thermochemical data, use various functionals to get an idea of the possible spread of results.
Closing remarks First-principles DFT calculations have evolved into a very powerful and versatile theoretical tool, useful for analysis and explanation or even prediction of a large variety of phenomena. However, one has to be aware of the principal limitations of the DFT approach and of the unavoidable approximations which enable us to do large-scale calculations for the systems of real interest. So far, this tool has high credibility. To maintain this high standard, the users ought to perform each individual project with utmost care. Please do all the necessary checks and convergence tests!